In first-passage percolation (FPP), random weights are placed on the edges of a graph and used to define a random metric t(x,y). On the d-dimensionsal integer lattice Z^d, many questions remain about the large-scale behavior of the metric and its geodesics. In the 1990s, C. Newman conjectured that (for d = 2) infinite geodesics should have asymptotic direction, and that geodesics having the same direction should merge. There is also a longstanding claim by physicists that Var(t(0,x)) should be smaller than |x|^{1 - epsilon}, and some progress towards this was made in special cases by Benjamini-Kalai-Schramm and Benaim-Rossignol. I will discuss my work on these and related questions, including a proof of a version of Newman's conjecture (that geodesics are directed in sectors) and a proof that the sublinear variance phenomenon holds for general distributions.